Consider the split exact sequence obtained by adjoining a unit to the -algebra . Define the scalar mapping to be In other words, for all and . Notice that , and that belongs to for each . Let be the star homomorphism induced by . The image of is the subset of consisting of all matrices with scalar entries, and again for all elements in the unitzation. An element in or will be called a scalar element if .
The scalar mapping is natural in the sense that if and are -algebras, and if is a star homomorphism, then we get a commuting diagram, where .